Thread: Epsilon delta definition of limit

1. Epsilon delta definition of limit

How to prove that $\lim_{x\to2}{x^2}=4$ by using the formal definition of limit?
I tried to solve it, but the problem was to find delta that depends on epsilon.

2. Consider:

$\left |{x^2-4}\right |< \epsilon \Leftrightarrow \left |{x-2}\right |\left |{x+2}\right |< \epsilon$

For choosing $\delta$ take into account that $|x+2|$ is (for example) bounded on $[1,3]$.

Regards.

Fernando Revilla

3. I am sorry, but I still couldnot get it.

4. On $(1,3)$ we have $|x+2|<5$,

then, on $(1,3)$

$|x^2-4|=|x-2||x+2|<5|x-2|<\epsilon \Leftrightarrow |x-2|<\epsilon/5$

Now, choose

$\delta=\min\{1,\epsilon/5\}$

Regards.

http://www.fernandorevilla.es/